Viola: Gruppi di permutazioni e risultati di irrazionalità

Viola, Carlo:
Gruppi di permutazioni e risultati di irrazionalità
Bollettino dell'Unione Matematica Italiana Serie 9 1 (2008), fasc. n.2, p. 375-400, (Italian)
pdf (471 Kb), djvu (221 Kb). | MR 2424300 | Zbl 1215.11073

Sunto

We recall some basic concepts in diophantine approximation, in particular the notion of irrationality measure. We describe the main aspects of the permutation group method due to G. Rhin and the author, with some arithmetical applications.

Referenze Bibliografiche

[1] F. AMOROSO - C. VIOLA, Approximation measures for logarithms of algebraic numbers, Ann. Scuola Norm. Sup. Pisa (4) 30 (2001), 225-249. | fulltext EuDML | MR 1882030 | Zbl 1008.11028

[2] R. APÉRY, Irrationalité de $\zeta(2)$ et $\zeta(3)$, Astérisque 61 (1979), 11-13.

[3] K. BALL - T. RIVOAL, Irrationalité d'une infinité de valeurs de la fonction zêta aux entiers impairs, Invent. Math. 146 (2001), 193-207. | fulltext (doi) | MR 1859021 | Zbl 1058.11051

[4] F. BEUKERS, A note on the irrationality of $\zeta(2)$ and $\zeta(3)$, Bull. London Math. Soc. 11 (1979), 268-272. | fulltext (doi) | MR 554391 | Zbl 0421.10023

[5] L. EULER, De fractionibus continuis dissertatio, Commentarii Acad. Sci. Imper. Petropol. 9 (1737), 98-137. Opera omnia, ser. I vol. 14, Teubner, Leipzig-Berlin, 1925, 187-215. | MR 232653

[6] S. FISCHLER, Groupes de Rhin-Viola et intégrales multiples, J. Théor. Nombres Bordeaux 15 (2003), 479-534. | fulltext EuDML | MR 2140865 | Zbl 1074.11040

[7] L. GIACARDI - C. S. ROERO - C. VIOLA, Sui contributi di Lagrange alla teoria dei numeri, Rend. Sem. Mat. Univ. Pol. Torino 53 (1995), 151-181. | MR 1452376

[8] G. H. HARDY - E. M. WRIGHT, An Introduction to the Theory of Numbers, IV ediz., Oxford Univ. Press, 1960. | MR 67125

[9] M. HATA, Rational approximations to π and some other numbers, Acta Arith. 63 (1993), 335-349. | fulltext EuDML | fulltext (doi) | MR 1218461 | Zbl 0776.11033

[10] M. HATA, Rational approximations to the dilogarithm, Trans. Amer. Math. Soc. 336 (1993), 363-387. | fulltext (doi) | MR 1147401 | Zbl 0768.11022

[11] H. RADEMACHER, Topics in Analytic Number Theory, Grundlehren math. Wiss., Band 169, Springer-Verlag, Berlin-Heidelberg, 1973. | MR 364103 | Zbl 0253.10002

[12] G. RHIN - C. VIOLA, On a permutation group related to $\zeta(2)$, Acta Arith. 77 (1996), 23- 56. | fulltext EuDML | fulltext (doi) | MR 1404975 | Zbl 0864.11037

[13] G. RHIN - C. VIOLA, The group structure for $\zeta(3)$, Acta Arith. 97 (2001), 269-293. | fulltext EuDML | fulltext (doi) | MR 1826005 | Zbl 1004.11042

[14] G. RHIN - C. VIOLA, The permutation group method for the dilogarithm, Ann. Scuola Norm. Sup. Pisa (5) 4 (2005), 389-437. | fulltext EuDML | MR 2185863 | Zbl 1170.11316

[15] G. RHIN - C. VIOLA, Multiple integrals and linear forms in zeta-values, Funct. Approx. Comment. Math. 37 (2007), 429-444. | fulltext (doi) | MR 2363836 | Zbl 1193.11073

[16] T. RIVOAL, La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs, C. R. Acad. Sci. Paris, Sér. I Math., 331 (2000), 267-270. | fulltext (doi) | MR 1787183 | Zbl 0973.11072

[17] K. F. ROTH, Rational approximations to algebraic numbers, Mathematika 2 (1955), 1-20. | fulltext (doi) | MR 72182 | Zbl 0064.28501

[18] C. L. SIEGEL, Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss., 1929, n. 1. Gesammelte Abhandlungen, Band I, Springer-Verlag, Berlin-Heidelberg, 1966, 209-266. | MR 197270

[19] C. VIOLA, Hypergeometric functions and irrationality measures, in: Analytic Number Theory, Y. Motohashi (ed.), London Math. Soc. Lecture Note Series 247, Cambridge Univ. Press, 1997, 353-360. | fulltext (doi) | MR 1695002 | Zbl 0904.11020

[20] C. VIOLA, Birational transformations and values of the Riemann zeta-function, J. Théor. Nombres Bordeaux 15 (2003), 561-592. | fulltext EuDML | MR 2140868 | Zbl 1074.11041

[21] C. VIOLA, The arithmetic of Euler's integrals, Riv. Mat. Univ. Parma (7) 3* (2004), 119-149. | MR 2128843 | Zbl 1166.11340

[22] C. VIOLA, Approssimazione diofantea, frazioni continue e misure d'irrazionalità, Boll. U.M.I. (8) 7-A (2004), 291-320. Addendum, Boll. U.M.I. (8) 8-A (2005), 179-182. | fulltext bdim | fulltext bdim | fulltext EuDML

[23] C. VIOLA, On the equivalence of Beukers-type and Sorokin-type multiple integrals, in: Proceedings of the conference "Diophantine and analytic problems in number theory", Moscow, 29/1 - 2/2/2007 (in corso di pubblicazione).

[24] W. ZUDILIN, Arithmetic of linear forms involving odd zeta values, J. Théor. Nombres Bordeaux 16 (2004), 251-291. | fulltext EuDML | MR 2145585 | Zbl 1156.11327

bdim: Biblioteca Digitale Italiana di Matematica

Un progetto SIMAI e UMI

Referenza completa

Sunto

Referenze Bibliografiche